MATEMATIQKI VESNIK
UDK 517.546
originalni nauqni rad
research paper
60 (2008), 207–213
A CLASS OF MULTIVALENT HARMONIC FUNCTIONS
INVOLVING A GENERALIZED RUSCHEWEYH TYPE OPERATOR
Waggas Galib Atshan, S. R. Kulkarni and R. K. Raina
Abstract. A class of p-valent harmonic functions associated with a certain generalized
Ruscheweyh type operator is introduced. Among the various properties investigated for this class
of functions are the results giving the coefficient bounds, distortion properties and extreme points.
1. Introduction
A continuous function f = u + iv is a complex valued harmonic function in
a complex domain C if both u and v are real harmonic in C. In any simplyconnected domain D ⊂ C, we can write f = h + g, where h and g are analytic in
D. We call h the analytic part and g the co-analytic part of f . A necessary and
sufficient condition for f to be locally univalent and sense-preserving in D is that
|h0 (z)| > |g 0 (z)| in D. See Clunie and Sheil-Small [3].
Denote by H(p) the class of functions f = h + g that are harmonic multivalent
and sense-preserving in the unit disk U = {z : |z| < 1}. For f = h + g ∈ H(p), we
may express the analytic functions h and g as
∞
∞
P
P
h(z) = z p +
an z n , g(z) =
bn z n , |bp | < 1.
(1.1)
n=p
n=p+1
Let W (p) denote the subclass of H(p) consisting of functions f = h + g, where h
and g are given by
∞
∞
P
P
h(z) = z p −
|an |z n , g(z) = −
|bn |z n , |bp | < 1.
(1.2)
n=p
n=p+1
We introduce here a new class Hλk (p, α, β) of harmonic functions of the form
(1.1) that satisfy the inequality
(
)
(Dλk+p−1 f (z))0
Dλk+p−1 f (z)
α
Re (1 − β)
+β
≥ ,
(1.3)
zp
pz p−1
p
AMS Subject Classification: 30C45
Keywords and phrases: Multivalent functions, harmonic functions, distortion bounds, extreme points, Ruscheweyh type operator.
207
208
W. G. Atshan, S. R. Kulkarni, R. K. Raina
where 0 ≤ α < p, p ∈ N = {1, 2, . . . }, λ ≥ 0, β ≥ 0, k ∈ N0 and
Dλk+p−1 f (z) = Dλk+p−1 h(z) + Dλk+p−1 g(z).
(1.4)
The operator Dλk+p−1 denotes the generalized Ruscheweyh derivative operator introduced in [2]. For h and g given by (1.1), we obtain
Dλk+p−1 h(z) = z p +
Dλk+p−1 g(z) =
∞
P
∞
P
(1 + λ(n − p))C(k, n, p)an z n ,
(1.5)
n=p+1
(1 + λ(n − p))C(k, n, p)bn z n ,
(1.6)
n=p
where λ ≥ 0, p ∈ N, k > −p and
µ
¶
n+k−1
C(k, n, p) =
.
k+p−1
(1.7)
We deem it worthwhile to point here the relevance of the function class
Hλk (p, α, β) with those classes of functions which have been studied recently. Indeed,
we observe that:
(i) H00 (1, α, 1) ≡ NH (α) (Ahuja and Jahangiri [1]);
(ii) Hλk (p, α, 1) ≡ Hλk (p, α) (Al Shaqsi and Darus [2]);
(iii) Hλk (1, 0, 1) ≡ Hλk (Darus and Al Shaqsi [4]);
∗
(Silverman [6]);
(iv) H00 (1, 0, 1) ≡ SH
(v) Hλ0 (1, 0, 1) ≡ H(λ) (Yalçin and Öztürk ][7]).
Also, we note that the analytic part of the class H0k (p, α, 1) was introduced
and studied by Goel and Sohi [5].
We further denote by Wλk (p, α, β) the subclass of Hλk (p, α, β) that satisfies the
relation
Wλk (p, α, β) = W (p) ∩ Hλk (p, α, β).
(1.8)
In this paper we study a class of p-valent harmonic functions involving a certain
generalized Ruscheweyh type operator. We obtain the coefficient bounds, distortion
properties and extreme points for this class of functions.
2. Coefficient bounds
Theorem 1. Let f = h + g (h and g being given by (1.1)). If
∞
P
n=p+1
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|an |+
+
∞
P
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|bn | ≤ p − α,
(2.1)
n=p
where λ ≥ 0, β ≥ 0, 0 ≤ α < p, p ∈ N and k ∈ N0 , then f is harmonic p-valent
sense-preserving in U and f ∈ Hλk (p, α, β).
209
Harmonic functions involving Ruscheweyh type operator
D k+p−1 f (z)
(D k+p−1 f (z))0
Proof. Let w(z) = (1 − β) λ zp
+ β λ pzp−1
. To prove that Re{w} ≥
α
p , it is sufficient to show equivalently that |p − α + pw(z)| ≥ |p + α − pw(z)|.
Substituting for w(z) and making use of (1.4) to (1.6), and resorting to simple
calculations, we find that
∞
P
|p − α + pw(z)| ≥ 2p − α −
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|an ||z n−p |
n=p+1
∞
P
−
n=p
and
∞
P
|p + α − pw(z)| ≤ α +
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|bn ||z n−p |
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|an ||z n−p |+
n=p+1
∞
P
+
n=p
(2.2)
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|bn ||z n−p |,
(2.3)
where C(k, n, p) is given by (1.7). Evidently, (2.2) and (2.3) in conjunction with
(2.1) yields
|p − α + pw(z)| − |p + α − pw(z)| ≥ 0.
The harmonic functions
∞
P
xn
f (z) = z p +
zn+
n=p+1 ((n − p)β + p)(1 + λ(n − p))C(k, n, p)
∞
P
yn
+
(z)n (2.4)
((n
−
p)β
+
p)(1
+
λ(n − p))C(k, n, p)
n=p
P∞
P∞
( n=p+1 |xn | + n=p |yn | = p − α) show that the coefficient bound given by (2.1)
is sharp.
The functions of the form (2.4) are in Hλk (p, α, β) because in view of (2.1), we
infer that
∞
P
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|an |+
n=p+1
∞
P
+
n=p
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|bn | =
∞
P
n=p+1
|xn | +
∞
P
n=p
|yn | = p − α.
The restriction imposed in Theorem 1 on the moduli of the coefficients of f = h + g
implies that for arbitrary rotation of the coefficients of f , the resulting functions
would still be harmonic multivalent and f ∈ Hλk (p, α, β).
The following theorem shows that the condition (2.1) is also necessary for
function f to belong to Wλk (p, α, β).
Theorem 2. Let f = h + g with h and g are given by (1.2). Then f ∈
Wλk (p, α, β) if and only if
∞
P
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|an |+
n=p+1
+
∞
P
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|bn | ≤ p − α,
n=p
(2.5)
210
W. G. Atshan, S. R. Kulkarni, R. K. Raina
where λ ≥ 0, β ≥ 0, 0 ≤ α < p, p ∈ N and k ∈ N0 .
Proof. By noting that Wλk (p, α, β) ⊂ Hλk (p, α, β), the sufficiency part of Theorem 2 follows at once from Theorem 1. To prove the necessary part, let us assume
that f ∈ Wλk (p, α, β). Using (1.3), we get
(
Ã
!
Dλk+p−1 h(z) + Dλk+p−1 g(z)
+
Re (1 − β)
zp
Ã
!)
(Dλk+p−1 h(z))0 + (Dλk+p−1 g(z))0
+β
pz p−1
(
∞
P
n
= Re 1 −
(( − 1)β + 1)(1 + λ(n − p))C(k, n, p)|an |z n−p −
n=p+1 p
)
∞
P
n
α
n−p
−
(( − 1)β + 1)(1 + λ(n − p))C(k, n, p)|bk |(z)
≥ .
p
n=p p
If we choose z to be real and let z → 1− , we obtain
∞
P
1−
((
n=p+1
n
− 1)β + 1)(1 + λ(n − p))C(k, n, p)|an |−
p
∞
P
n
α
−
(( − 1)β + 1)(1 + λ(n − p))C(k, n, p)|bn | ≥ .
p
n=p p
Hence
∞
P
n=p+1
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|an |+
+
∞
P
n=p
((n − p)β + p)(1 + λ(n − p))C(k, n, p)|bn | ≤ p − α,
which completes the proof of Theorem 2.
3. Distortion bounds and extreme points
In this section we obtain the distortion bounds for functions belonging to the
class Wλk (p, α, β) and also provide extreme points for this class Wλk (p, α, β).
Theorem 3. If f ∈ Wλk (p, α, β), for λ ≥ 0, β ≥ 0, 0 ≤ α < p, p ∈ N, k ∈ N0
and |z| = r > 1, then
|f (z)| ≤ (1 + |bp |)rp +
(p − α) − |bp |
rp+1 ,
(β + p)(λ + 1)(p + k)
(3.1)
|f (z)| ≥ (1 − |bp |)rp −
(p − α) − |bp |
rp+1 .
(β + p)(λ + 1)(p + k)
(3.2)
and
211
Harmonic functions involving Ruscheweyh type operator
Proof. We only prove the first inequality (3.1). The proof for the second
inequality (3.2) is similar, and is hence omitted.
Suppose f ∈ Wλk (p, α, β). Using (1.1) and (2.1) of Theorem 1, we find that
|f (z)| ≤ (1 + |bp |)rp +
= (1 + |bp |)rp +
∞
P
n=p+1
n=p+1
n=p+1
(|an | + |bn |)rn ≤ (1 + |bp |)rp +
∞
P
(|an | + |bn |)rp+1
n=p+1
1
×
(β + p)(1 + λ)(p + k)
(β + p)(1 + λ)(p + k)(|an | + |bn |)rp+1
≤ (1 + |bp |)rp +
∞
P
∞
P
1
×
(β + p)(1 + λ)(p + k)
((n − p)β + p)(1 + λ(n − p))C(k, n, p)(|an | + bn |)rp+1
≤ (1 + |bp |)rp +
1
[(p − α) − |bp |]rp+1 .
(β + p)(1 + λ)(p + k)
The bounds given in Theorem 3 ( for the functions f = h + g of the form (1.2))
also hold for functions of the form (1.1) if the coefficient condition (2.1) is satisfied.
The functions
f (z) = z p + |bp |(z)p +
(p − α) − |bp |
(z)p+1
(β + p)(1 + λ)(p + k)
(3.3)
f (z) = z p − |bp |(z)p −
(p − α) − |bp |
(z)p+1
(β + p)(1 + λ)(p + k)
(3.4)
and
for |bp | < 1 show that the bounds given in Theorem 3 are sharp.
The covering result given below in Corollary 1 follows from the inequality (3.2)
of Theorem 3.
Corollary 1. If f ∈ Wλk (p, α, β), then
½
¾
(p − α) − |bp |
w : |w| < (1 − |bp |) −
⊂ f (U).
(β + p)(λ + 1)(k + p)
(3.5)
The next theorem gives the extreme points of the closed convex hulls of
Wλk (p, α, β), denoted by clcoWλk (p, α, β)
Theorem 4. f ∈ clcoWλk (p, α, β) if and only if
f (z) =
∞
P
n=p
(σn hn + En gn ),
(3.6)
212
W. G. Atshan, S. R. Kulkarni, R. K. Raina
where z ∈ U , hp (z) = z p ,
hn (z) = z p −
p−α
zn,
((n − p)β + p)(1 + λ(n − p))C(k, n, p)
gn (z) = z p −
p−α
(z)n ,
((n − p)β + p)(1 + λ(n − p))C(k, n, p)
and
∞
P
n=p
(n = p + 1, p + 2 . . . )
(3.7)
(n = p, p + 1, . . . )
(3.8)
(σn + En ) = 1(σn ≥ 0, En ≥ 0).
In particular, the extreme points of Wλk (p, α, β) are {hn } and {gn }.
Proof. Suppose f (z) is of the form (3.6). Using (3.7) and (3.8), we get
∞
P
f (z) =
(σn hn + En gn )
n=p
∞
P
∞
P
p−α
σn z n
((n
−
p)β
+
p)(1
+
λ(n
−
p))C(k,
n,
p)
n=p+1
∞
P
p−α
−
En (z)n
((n
−
p)β
+
p)(1
+ λ(n − p))C(k, n, p)
n=p
∞
P
p−α
σn z n
= zp −
((n
−
p)β
+
p)(1
+ λ(n − p))C(k, n, p)
n=p+1
∞
P
p−α
−
En (z)n .
((n
−
p)β
+
p)(1
+ λ(n − p))C(k, n, p)
n=p
=
n=p
Then
∞
P
(σn + En )z p −
p−α
σn
((n − p)β + p)(1 + λ(n − p))C(k, n, p)
n=p+1
∞
P
p−α
+
[((n−p)β +p)(1+λ(n−p))C(k, n, p)]
En
((n − p)β + p)(1 + λ(n − p))C(k, n, p)
n=p
Ã
!
∞
P
(σn + En ) − σp = (p − α)(1 − σp ) ≤ p − α
= (p − α)
[((n−p)β+p)(1+λ(n−p))C(k, n, p)]
n=p
which implies that f ∈ clcoWλk (p, α, β).
Conversely, assume that f ∈ Wλk (p, α, β). Putting
((n − p)β + p)(1 + λ(n − p))C(k, n, p)
|an | (n = p + 1, p + 2, . . . ),
p−α
((n − p)β + p)(1 + λ(n − p))C(k, n, p)
En =
|bn | (n = p, p + 1, p + 2, . . . ),
p−α
we get
∞
X
(σn hn + En gn ),
f (z) =
σn =
n=p
and this completes the proof of Theorem 4.
Harmonic functions involving Ruscheweyh type operator
213
REFERENCES
[1] O. Ahuja and J. Jahangiri, Noshiro-type harmonic univalent functions, Sci. Math. Japan,
65 (2002), 293–299.
[2] K. Al Shaqsi and M. Darus, A new class of multivalent harmonic functions, General Math.,
14 (2006), 37–46.
[3] J. Clunie and T. Shell-Small, Harmonic univalent functions, Ann. Acad. Aci. Fenn. Ser. A
I Math., 9 (1984), 3–25.
[4] M. Darus and K. Al Shaqsi, On harmonic univalent functions defined by a generalized
Ruscheweyh derivatives operator, Lobachevskii J. Math., 22 (2006), 19–26.
[5] R. Goel and N. Sohi, New criteria for p-valence, Indian J. Pure Appl. Math., 7 (2004), 55–61.
[6] H. Silverman, Harmonic univalent functions with negative coefficient, Proc. Amer. Math.
Soc., 51 (1998), 283–289.
[7] S. Yalçin and M. Öztürk, A new subclass of complex harmonic functions, Math. Inequal.
Appl., 7 (2004), 55–61.
(received 15.08.2007)
Waggas Galib Atshan, Department of Mathematics, College of Computer Science and Mathematics, Univesrity of AL Qadisiya, Diwaniya, Iraq
E-mail: waggashnd@yahoo.com
S. R. Kulkarni, Department of Mathematics, Fergusson College, Pune – 411004, India
E-mail: kulkarni ferg@yahoo.com
R. K. Raina, 10/11, Ganpati Vihar, Opposite Sector 5, Udaipur 313002, Rajasthan, India
E-mail: rainark 7@hotmail.com